Higher-order variation and pathwise Ito calculus on manifolds
Abstract
We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a p-th variation tensor along a sequence of partitions, for an arbitrary integer p; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite p-th variation and derive a change of variable formula for smooth functions of such paths. For p=2, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced p-jet of the test function, whose canonical highest-order component is determined by the p-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.
Related
- Higher-Order Paracontrolled Calculus for Geometric Rough Paths
- Jet Bundles as Higher-Order Polarised k-Contact Manifolds
- Extending Pathwise Gradients to Discrete Random Variables via Finite-Order Relaxation
- Differential calculus on quantized irreducible flag manifolds
- Sparse domination for q-variation associated with functional calculus
BibTeX
@misc{cont2026higherorder,
title = {{Higher-order variation and pathwise Ito calculus on manifolds}},
author = {Rama Cont},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.12225},
eprint = {2608.12225},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12225},
}
RIS
TY - UNPB
TI - Higher-order variation and pathwise Ito calculus on manifolds
AU - Rama Cont
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.12225
UR - https://arxiv.org/abs/2608.12225
AB - We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a p-th variation tensor along a sequence of partitions, for an arbitrary integer p; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite p-th variation and derive a change of variable formula for smooth functions of such paths. For p=2, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced p-jet of the test function, whose canonical highest-order component is determined by the p-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.
ER -